Comparative Analysis of Fourth-Order Convergence Numerical Methods Applied to Geometric Nonlinear Problems in Planar Trusses Using the Corotational Finite Element Method

Resumo

This article presents a comparative analysis of iterative methods applied to solving nonlinear systems of equations arising from the structural analysis of planar trusses with geometric nonlinearity. The study investigates the classical Newton-Raphson method, which has quadratic convergence, and two fourth-order methods: Soleymani's method and the optimized Potra-Pták (OPP) method, the latter proposed by Cordero et al. All algorithms are implemented in the open-source software Scilab and are integrated with continuation techniques to trace the equilibrium path, including limit points. The structures are modeled using the Corotational Finite Element Method formulation. The evaluation of the methods considers criteria such as computational efficiency, total number of accumulated iterations, and robustness in obtaining solutions near limit points of force or displacement. The results show that, although the Newton-Raphson method stands out for its simplicity and general applicability, higher-order methods, such as those of Soleymani and OPP, exhibit superior performance in terms of convergence speed and numerical stability, especially in analyses involving large displacements. The study highlights the potential of high-order methods to improve the efficiency of nonlinear structural simulations.